Alternative models of the sound field in a reverberant room

t.t'r¡

ALTERNAÎIVE MODELS OF lHE ^SOTJND FIEIÐ

IN

A REVERBERAI{T ROOM

by

Thomas

J. Munro

B. Sc.

A thesÍs

presented

to the

Faeulty

of

Engineering

of the Unfverslty of

Adelalde

for the

Degree

of

Master

of

Englneering Sclence

Department

of

Mechanfcal Englneering

University of

^Adelaide

January 1982

TABLE OF CONTENTS

Summary

Statement

of

^{0ri gì naì i}

ty

Acknowl edgements

chapter

I

^AC0USTIC MEASUREMENTS IN REVERBERATI0N R00MS

l.l

Introduct'ion

1.2

tory of

Reverberant Sound Measurements

1'l

1V

vi

vìi

I I 3

4 9 12

Ear'ly Ì¡lork

'in

_{Room Acoustics}

The

Statistical Variation of

^Measured

Quantì t'ies

Determ'i nat'ion

of

Sound Power I

niecti

on i n

Reverberant F'iel ds

Methods

of

Obtainingan Accurate Space Average

of

^l4easured Quantì ^b'ies

Stati

sti cal

Vari a'b'ion ì

n

Decayi ng Sound Fi el ds Comparison

of Stat'ist'ical

^and Modal Models

of

Room Behavi our

Predict'ion

of

Rate

of

^{Decay by}

Aìternative

Model s

1.3

^Standard l4easurenent Methods Power Level

Absorption Coeff

ici

ent Transmi ss'ion Loss nd

nd nd

His 1.2 1.2

| .2.5

1 .2,'o

1.3.1

r.3.2

I .3.3

2

1 .2.3

1 .2.4

1 .2.7

l4 l5

Sou Sou Sou

17

l8

20 21 25 27

Chapter ² 2.1

2.6 Chapter 3

3.1 3.2 2.2 2.3

ALTERNATIVE THEORETICAL DEVELOPMENTS I ntroduct i on

The hlave Equation and Soìutions

Steady

State

^Behaviour

of the

^{Modal Sound}

Field

2 .3

.1

Equal Energy Di s'lri ^{buti on} 2.3

.2

^{Equaì Power F I ow}

Modal Descri

pt'ion of the

Decayi ng Sound F'iel d

Modal Damping by

the

Acoust'ic Ray Approxitnalion

2.5.1

^Predi

ction of

^{Modaì Decay Rate}

2.5.2 Norris-Eyrinq

Equation as

a

Spec'ial ^Case M'iscel I aneous Formul as

MATERIALS AND METHODS

Phys'icaì Fac'i I

it'ies

Instrumentat'ion, ^Sì gnaì Generat'ion and Measurement

3.2.1

Generation

of

^Sound

3.2.2

Measurement and Process'ing

of

^{Sounci Pressure}

Level s

Experìmental Methods

3.3..l

Measurement

of

^{Modal Decay Rate}

3.3.2

Measurement

of

Steady

State

Modal Energy Level

37 30 30 32

37 40

41

43 43 48

5l 5l

53 54 56 57 57 60

?.4 2.5

49

3.3

Chapter ⁴ 4.1

4.2 4.3

4.4 Chapter ⁵

5.1 5.2 Chapter ⁶ References

4.3. I

4.3.2

RESULTS: MODAL BEHAVIOUR IN THE EMPTY ROOM

Veri f i cati on

of

l4odal Behav'iour

4.1.1

Modal Behaviour

in

Steady

State

^Conditions

4.1

.2

^{Modal Behavi}

our

^ì

n a

^{Decay'i ng} Sound Fi el d

The

llature of

^{Moda'l DecaY}

Anaìysì

s of the

Modal Rate

of

^DecaY

4.3.3

The

Tradìtional

ltlodal ^Approach Modal Damping by

the

RaY Tracing Approxi rnatì on

Probable Reasons

for the Fa'ilure of the

Modeìs

to

Pred'ict

Actual

Behaviour

Initial

^Energy

Level of

^{Room Modes}

THE ROTATING DIFFUSER, ITS EFFECT ON MODAL STRUCTURE

Previous l,Jork

lvith

Rotat'ing

Diffusers

Experimental Results

CONCLUS IONS

lll

Page

62 62 62 63 68 73 77

79 83 84

90 9t

92

l0l

103

]V

SUMMARY

Current standard test

^methods

for acoustic

measurements

nade

in

reverberant

^rooms

have

been

found to y'ieìd inconsìstent results at

^lolv

frequencies. The theoret'ical basis of the standard test

^methods

relìes

upon

the

assumption

that the reverberant

sound

field can be

^{modeìed as}

compìeteìy d'iffuse ab

^al

I

^f

requencies. Thus,

^'inconsi

stenc'ies have

^been

attributed to a lack of diffusivity at low frequencies. Th'is

thesis

exami

nes the

^appropri

ateness of the

^di

ffuse fi

^el

d

^assumpti

on at

^{I ow}

frequencìes and

shows

that the strong modal characteristics of a

^low

frequency reverberant

sound

field precìudes the possibi'lìty of

^mode'ling

such a

field

^as

dìffuse.

A

nlodel

of the reverberant

^sound

field which includes the

^observed

modal characteri

st'ics and

ⁱ

s

based upon wel

I

known sol

ut'ions to the

^wave

equa'ti

on

i

s

^presented

.

^{Th j}

s

^{model 'i}

s

dependent

on the

^{accurate predi}

cti

^on

of the steady state

^ampì

'itude

^and

the

^decay

rate of

^{i nd'ivi}

dual

^modes

.

^{I n}

pred'icti

ng the steady state

^ampì

'itude of the

^{modes contri}

bui'ing to

^a

reverberant

sound

f

iel

d, the

assumpt'ion i

s

usual

ly

^made

e'ither

^{i mpì i}

c'itìy or expììcìt1y that there 'is equal

_energy

distributioç

between modes. In th'i

s

^thes'i

s, an

^al

ternati ve

approach based

on the

assumpti

on that

^there

'is

_{equal power}

flow to alì

^modes

vlill also

be consìdered. These assumpt'ions

are

di scussed and

the'ir

^{i nf I} uence

on the

^{rel ati}

ve

di

ff

erence ^'i n

the

steady

state

^sound

pressure

^I

evel of

^{any two modes i}

s

^{analyti ca'l}

ly

^'i nvest'igated.

Two models

for predicting the decay rate of ind'iv'idual

modes are

also

examined. The

fjrst is

^{based upon}

an

approx'imate

solution to the

^wave

equation for a very ìightìy

^{damped room.} The second

is

based upon

a

^com-

b'ina'li

on of the modal

approach

and the ray tracì ng

approach

as

appì 'ied

to the rectangu'lar room. Both

models

reìy

^upon

the

assumption

that

the

wal I

s can be accurateìy

^model

ed as

^'loca'l

ly

^reacti

ve and that the

^{wal l}

V

.impedance can be modeled

as constant over the wall

^{surfaces and throughout}

the

frequency range

investigated.

Experintental results are presented for

comparison

of each

^model.

Results ind'icate that the wall surfaces of the

room cannot be modeled ^as

being locally

^reac'bive

as theìr

^response

to excitation by the

reverberant sound

fjeld in the

^frequency

range'investjgated is characterized by

^modaì

vibration. In

consequence

ìt

^j

s

^shown

that neìther model

accurate'ly

pred'icts the relative rates of

^decay

of the

^measured modes. The

relative

ampì i

tudes of

^{i nd'ivi}

dual

^modes

are seen to be h'igh'ly

^dependent

on

the

effective coupìing

between each mode and

the

sound

source.

Th'is

is

^shown

to

^be

highly

unpredictable

for

^commonly used sound sources.

A

^wi

deìy

recommended means

of

enhanc'ing

the

^di

ff

usì

on

^'i

n a

^rever-

berant room is the installation of a rotating diffuser. Th'is

^thes'is

examines

the effects of a rotatìng diffuser on the

modal

characteristics

of a reverberant

sound

field

^{and shows}

that jt substantially disrupts

^the nodal

structure of the

sound

field.

vl

To

the best of the candidate's

^{knowledge and}

belief'

thi s

^thesì

s

contai

ns no

^materi

al

^whi

ch has

^been

accepted

for the

award

of

any degree

or

^dipìoma

in

^any

Unì versi

ty, and

^conta'i

ns no

^materi

al

previ ously

pubìished or written by another person, except

^where due reference

'is

_made

'in the text.

Thomas

J.

^Munro

vii

ACKNOt.lLEDGEMENTS

The

work

described

in this thesis

^was

carried out in the

Departntent

of

^Mechani

cal

^{Engi neerì}

ng of the Un'ivers'ity of

^Adel

a'ide

under

the

^{aeg'i s}

of

^Prof

essor

^{R . E}

. Luxton.

^The

autholi s

^{'i ndebted}

to

Professor Luxton f or

the opportunity to carry out th'is

research.

To Dr

^{D .}

A.

^{B'i es}

, who

^supervi

sed the

^work

, the author

ⁱ

s

^deepìy

'indebted, both for his constant interest

^and encouragement throughout ^the course

of the

research and

for his helpful

suggest'ions

during the

prepara-

ti

on

of

^th'i

s

^{thesi s .}

To the University staff, thanks are

^due

for their assistance jn

_the

preparati

on of the

^{f aci I}

'iti es

i

n

wh'ich

the work was

^compì

eted, and

^{i n}

part'icular to

^Mr

H.

Bode and Mr

R.

Curt'in.

The author gratefully

acknowledges

the Australian

American Eciuca-

ti onal

^Foundati

on who

^provi

ded the

^{f i}

nanc'ial support wh'ich

^{made the} conrpletion

of this

work possible.